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Functional data analysis has significantly enriched the landscape of existing data analysis methodologies, providing a new framework for comprehending data structures and extracting valuable insights. This paper is dedicated to addressing functional data clustering—a pivotal challenge within functional data analysis. Our contribution to this field manifests through the introduction of innovative clustering methodologies tailored specifically to functional curves. Initially, we present a proximity measure algorithm designed for functional curve clustering. This innovative clustering approach offers the flexibility to redefine measurement points on continuous functions, adapting to either equidistant or nonuniform arrangements, as dictated by the demands of the proximity measure. Central to this method is the “proximity threshold”, a critical parameter that governs the cluster count, and its selection is thoroughly explored. Subsequently, we propose a time-shift clustering algorithm designed for time-series data. This approach identifies historical data segments that share patterns similar to those observed in the present. To evaluate the effectiveness of our methodologies, we conduct comparisons with the classic K-means clustering method and apply them to simulated data, yielding encouraging simulation results. Moving beyond simulation, we apply the proposed proximity measure algorithm to COVID-19 data, yielding notable clustering accuracy. Additionally, the time-shift clustering algorithm is employed to analyse NASDAQ Composite data, successfully revealing underlying economic cycles.

Functional data analysis has significantly enriched the landscape of existing data analysis methodologies, providing a new framework for comprehending data structures and extracting valuable insights. This paper is dedicated to addressing functional data clustering—a pivotal challenge within functional data analysis. Our contribution to this field manifests through the introduction of innovative clustering methodologies tailored specifically to functional curves. Initially, we present a proximity measure algorithm designed for functional curve clustering. This innovative clustering approach offers the flexibility to redefine measurement points on continuous functions, adapting to either equidistant or nonuniform arrangements, as dictated by the demands of the proximity measure. Central to this method is the "proximity threshold", a critical parameter that governs the cluster count, and its selection is thoroughly explored. Subsequently, we propose a time-shift clustering algorithm designed for time-series data. This approach identifies historical data segments that share patterns similar to those observed in the present. To evaluate the effectiveness of our methodologies, we conduct comparisons with the classic K-means clustering method and apply them to simulated data, yielding encouraging simulation results. Moving beyond simulation, we apply the proposed proximity measure algorithm to COVID-19 data, yielding notable clustering accuracy. Additionally, the time-shift clustering algorithm is employed to analyse NASDAQ Composite data, successfully revealing underlying economic cycles. [ABSTRACT FROM AUTHOR]

This paper presents a method for the digital history of a discipline (social psychology in this application) through the analysis of scientific publications. The titles of a comprehensive set of papers published in the Journal of Personality and Social Psychology (1965–2021) were collected, yielding a total of 10,222 items. The corpus thus constructed underwent several stages of preprocessing until the final conversion into a terms x time-points matrix, where terms are stemmed words and multi-words. After normalizing frequencies via a chi square-like transformation, clusters of words portraying similar temporal patterns were identified by functional (textual) data analysis and distance-based curve clustering. Among the best candidates in terms of the number of clusters, the solutions with six, nine and thirteen clusters (from lower to higher resolution) have been chosen and the nesting relationship demonstrated. They reveal—at different levels of granularity—increasing, decreasing, and stable keywords trends, highlighting methods, theories, and application domains that have become more popular in recent years, lost popularity, or have remained in common use. Moreover, this method allows to highlight historical issues (such as crises in the discipline or debates over the use of terms). The results highlight the core topics of social psychology in the past and today, underlying the crucial contribution of this method for the digital history of a discipline. [ABSTRACT FROM AUTHOR]

A model involving Gaussian processes (GPs) is introduced to simultaneously handle multitask learning, clustering, and prediction for multiple functional data. This procedure acts as a model-based clustering method for functional data as well as a learning step for subsequent predictions for new tasks. The model is instantiated as a mixture of multi-task GPs with common mean processes. A variational EM algorithm is derived for dealing with the optimisation of the hyper-parameters along with the hyper-posteriors' estimation of latent variables and processes. We establish explicit formulas for integrating the mean processes and the latent clustering variables within a predictive distribution, accounting for uncertainty in both aspects. This distribution is defined as a mixture of cluster-specific GP predictions, which enhances the performance when dealing with group-structured data. The model handles irregular grids of observations and offers different hypotheses on the covariance structure for sharing additional information across tasks. The performances on both clustering and prediction tasks are assessed through various simulated scenarios and real data sets. The overall algorithm, called MAGMACLUST, is publicly available as an R package. [ABSTRACT FROM AUTHOR]

This paper proposes a novel unsupervised clustering framework to identify driving style not in terms of the discrete features of driving behavior data, but rather the time-varying patterns of driving maneuver intensity. This framework can describe the dynamic decision-making process of driving behavior and the continuity of driving data, and driving maneuver intensity is the basic of this paper. Therefore, detection, feature analysis and clustered on intensity of driving maneuvers are carried out using a threshold-based approach, hierarchical feature extraction, and k-means clustering. Then, to analyze fine-grained driving style, dynamic time windows are determined according to road alignment. In dynamic time windows, this paper constructs time-varying patterns based on driving maneuver intensity, which consider the intensity and frequency of driving behavior and preserve the time-varying characteristics of time-series data. However, not all dynamic time windows are equal in maneuvers’ duration and number, which means the time-varying patterns of driving maneuver intensity are curves with various lengths. So that, for clustering time-varying patterns, this paper proposes a novel curve clustering algorithm named Similarity-Based Clustering with Dynamic Time Warping (SBC-DTW) that can cluster curves with various lengths. The empirical results based on real driving data demonstrate that the proposed framework can classify driving style more accurately than the classical method. Moreover, according to this framework, we can have an in-depth understanding of dynamic driving behavior and the composition of drivers’ long-term driving styles.

Hard landing is a typical flight safety incident, and interpretability plays an important role in flight safety research. However, existing studies failed to provide good interpretability of the reasons for hard landing incidents and suffer from low prediction accuracy. To address the above problems, in this paper we propose CurveCluster+, a curve clustering method based on quick access recorder (QAR) data for hard landing risk evaluation. Specifically, we first conduct an in-depth analysis on hard landing flights by comparing key QAR parameter curves with the group behavior, based on which we establish a two-level hierarchical classification of hard landing incidents according to the hard landing patterns. Then we extract curve-level features from key QAR parameters through interpolation and resampling. After that we turn the classic K-means clustering into a semi-supervised algorithm by incorporating some expert experience and apply it on the curve-level features to automatically recognize the hard landing patterns. Finally, we propose a risk evaluation model based on the clustering results to discover high-risk flights from normal ones. We evaluate our method on a QAR dataset of 37,943 Airbus 320 aircraft flights. The results show that compared with other state-of-the-art data-driven methods, CurveCluster+ provides strong interpretability of hard landing incidents and exhibits good performance in recognizing hard landing patterns (the overall accuracy of our method reaches up to 92.99%). Moreover, it only requires a handful of hard landing samples to discover high-risk flights from tremendous normal landing flights, which is critical for flight safety warnings. [ABSTRACT FROM AUTHOR]

Grid nanoindentation method is commonly adopted to probe micro-mechanical properties of composite materials, providing the basis of multi-scale design of materials and furtherly supporting the research of Materials Genome Engineering. To overcome the disadvantages of traditional nanoindentation data processing methods, a new analysis framework based on intelligence recognition technology was proposed in present work. Curve clustering analysis was established based on principal component analysis to identify the minerals of cementitious materials represented by the indentation load-depth curves. This method avoids the Gaussian distribution assumption and matches every mineral with the test data at effective indentation depth. Compared with deconvolution analysis, Gaussian Mixture Model and point clustering analysis, curve clustering analysis offers more reasonable results. Furthermore, two supervised classification methods, Support Vector Machine classifier and Back Propagation neural network classifier, are introduced to recognize and group nanoindentation data. The training and testing results indicate that they are feasible and highly accurate. • A new analysis framework based on intelligence recognition method is developed to deal with grid nanoindentation test data. • The curve clustering method can distinguish major mineral phases in higher accuracy compared with deconvolution analysis. • The effective indentation depth range of each phase can be determined by the derivative curve of average E - h curve. • Supervised classification methods perform well in the classification of grid nanoindentation test data with high accuracy. [ABSTRACT FROM AUTHOR]

• Literature of curve clustering is classified by similarity characteristics. • A new nomenclature is suggested in hope of easing the understanding of practitioners. • Good perspective of curve clustering problem by focusing on similarity characteristic. • Better insight on which method to use depending on the similarity characteristics. This paper presents an overview of curve clustering from the similarity characteristics perspective, with a goal of providing useful advice and references regarding fundamental concepts that are accessible to the broad community of curve clustering practitioners. We introduce a new taxonomy of curve clustering by proposing four major similarity characteristics. We reviewed some contributions to curve clustering with respect to their similarity characteristics along with their applications. Lastly, we give an in-depth discussion of the overall challenges in this field with respect to similarity characteristics, highlight open research questions and discuss guidelines for further progress. [ABSTRACT FROM AUTHOR]

The clustering for functional data with misaligned problems has drawn much attention in the last decade. Most methods do the clustering after those functional data being registered and there has been little research using both functional and scalar variables. In this article, we propose a simultaneous registration and clustering model via two-level models, allowing the use of both types of variables and also allowing simultaneous registration and clustering. For the data collected from subjects in different groups, a Gaussian process functional regression model with time warping is used as the first level model; an allocation model depending on scalar variables is used as the second level model providing further information over the groups. The former carries out registration and modeling for the multidimensional functional data (two-dimensional curves) at the same time. This methodology is implemented using an EM algorithm, and is examined on both simulated data and real data. for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper, a revised directional triangle-area curve representation method (DTAR) is proposed to address the problem of eyebrow semantic shape characterization via curve representation. First, 11 or 12 DTAR values are selected to describe eyebrows via considering the eyebrow corner information roughly, and then the corresponding DTAR curves are acquired via the cubic spline interpolation based on these selected points. Second, a descriptor of the landmarks is developed to represent selected reference eyebrows, and the corresponding DTAR curves are obtained for the selected reference eyebrows. Lastly, a similarity notion based on AFS is introduced via measuring the membership degrees of each eyebrow shape similar to the given reference shapes, and then one can describe each eyebrow shape by using two given reference eyebrow shapes via computing the membership degrees representing the relative similarities. To illustrate the effectiveness of the proposed approach, we use the AR and BJUT databases for experiments to demonstrate the consistency comparison with human perceptions. The experimental results show that the extracted semantic notions of eyebrow shapes obtained by the proposed approach are much better than those by only utilizing 11 DTAR values or 12 DTAR values directly in terms of the consistency with human perceptions. [ABSTRACT FROM AUTHOR]

• The comp utat ional effi cien cy of the optimization algorithm is improved for the MGPFR model. • The local maxi mum problem of the optimization algorithm is overcome for the mix-GPFR model. • The comp utat ional effi cien cy of the optimization algorithm is improved for the TMGPFR model. • The local maxi mum prob lem of the optimization algorithm is overcome for TMGPFR. The mixture of Gaussian processes is effective for regression, but it cannot handle the non-stationary curve clustering problem well. The two-layer mixture of Gaussian process functional regressions (TMGPFR) model was established to deal with this problem. In this paper, we first propose the classification EM (CEM) algorithm to solve that the optimization algorithm is inefficient for TMGPFRs, and then propose the deterministic annealing CEM algorithm for TMGPFRs to overcome the local maximum problem of the CEM algorithm. Lastly, experiments are conducted on synthetic and real-world data sets, and the results show that our proposed algorithms are more effective than the compared algorithms on curve clustering and regression. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel approach to address the problem of eye shape characterization via curve representation. Firstly, a directional triangle-area curve representation method (DTAR) is presented for this aim. Equipped with DTAR, the shape similarities between two eyes can be measured by the similarities between two corresponding DTAR curves. Secondly, in order to exploit the underlying information of eye shapes, a curve clustering algorithm is utilized to automatically discover a set of eye shape prototypes. Consequently, a semantics extraction method for eye shapes is proposed in terms of seven reference eye shapes. Finally, in order to validate the consistency of the clustering results and the extracted semantics, extensive experiments on AR and BU4DFE databases are designed and conducted, and all the results demonstrate the effectiveness of the proposed DTAR curve representation and the semantics extraction method. [ABSTRACT FROM AUTHOR]

Many data collected in sport science come from time dependent phenomenon. This article focuses on Functional Data Analysis (FDA), which study longitudinal data by modelling them as continuous functions. After a brief review of several FDA methods, some useful practical tools such as Functional Principal Component Analysis (FPCA) or functional clustering algorithms are presented and compared on simulated data. Finally, the problem of the detection of promising young swimmers is addressed through a curve clustering procedure on a real data set of performance progression curves. This study reveals that the fastest improvement of young swimmers generally appears before 16 years old. Moreover, several patterns of improvement are identified and the functional clustering procedure provides a useful detection tool. [ABSTRACT FROM AUTHOR]

The mixture of Gaussian processes is a powerful statistical learning model that can be effectively applied to curve clustering and prediction. However, the corresponding model selection problem, that is, selecting an appropriate number of components in the mixture, is rather difficult to solve. In our previous work, we established the split-and-merge automatic model selection algorithm for mixtures of Gaussian processes along the output space under the framework of Reversible Jump Markov Chain Monte Carlo (RJMCMC), which can not only determine the number of actual Gaussian processes but also dynamically adjust the Gaussian process components to avoid dependence on parameter initialization and initial partitioning of the dataset during the parameter learning on a given dataset. In this study, we propose two algorithms: Penalized Likelihood RJMCMC and Penalized Prior RJMCMC. The former integrates a penalized term into the likelihood, while the latter incorporates a penalized term into the prior and operates within the full Bayesian inference framework, both aiming to focus more sharply on determining the number of components in the convergence process. Furthermore, we prove the geometric ergodicity of the RJMCMC algorithm for the mixture of Gaussian processes model, ensuring convergence of the posterior distribution with sufficient iterations. The experimental results further demonstrate the robustness of our PP-RJMCMC algorithm in model selection, showing superior performance compared to traditional approaches in curve classification and clustering. Additionally, the prediction performance is comparable to the EM algorithm. Although not directly explored in this study, the RJMCMC results can be used to initialize the EM algorithm, which could potentially improve prediction accuracy and accelerate computation. • Penalized likelihood and simulated annealing are used in RJMCMC for mix-GP to focus on a component. • Penalty term is integrated into the prior for full Bayesian inference in RJMCMC. • NUTS sampler replaces HMC for adaptive tuning, boosting RJMCMC efficiency. • Ergodicity of RJMCMC for mix-GP has been proved, implying convergence. [ABSTRACT FROM AUTHOR]

The growth curve clustering problem is analyzed and its connection with the spectral relaxation method is described. For a given set of growth curves and similarity function, a similarity matrix is defined, from which the corresponding similarity graph is constructed. It is shown that a nearly optimal growth curve partition can be obtained from the eigendecomposition of a specific matrix associated with a similarity graph. The results are illustrated and analyzed on the set of synthetically generated growth curves. One real-world problem is also given. [ABSTRACT FROM AUTHOR]

The electricity load data recorded by smart meters contain plenty of knowledge that contributes to obtaining load patterns and consumer categories. Generally, the daily load curves are clustered first in order to obtain load patterns of each consumer. However, due to the volume and high dimensions of load curves, existing clustering algorithms are not appropriate in this situation. Thus, a fused load curve clustering algorithm based on wavelet transform (FCCWT) is proposed to solve this problem. The algorithm includes two main phases. First, FCCWT applies multilevel discrete wavelet transform (DWT) to convert the daily load curves for dimensionality reduction. Second, it detects clusters at two outputs of the first phase, and then fuses two groups of clusters with a subalgorithm named cluster fusion to achieve the optimized clusters. FCCWT is implemented on datasets of both China and United States. Their clustering performances are evaluated by diverse validity indices comparing with four typical clustering methods. The experimental results show that FCCWT outperforms other comparison methods. Additionally, case analysis of two datasets are also provided to discuss the significance of load patterns. [ABSTRACT FROM PUBLISHER]

The growing availability of large diachronic corpora of scientific literature offers the opportunity of reading the temporal evolution of concepts, methods and applications, i.e., the history of disciplines involved in the strand under investigation. After a retrieval process of the most relevant keywords, bag-of-words approaches produce words  ×  time-points contingency tables, i.e. the frequencies of each word in the set of texts grouped by time-points. Through the analysis of word counts over the observed period of time, main purpose of the study is, after reconstructing the “life-cycle” of words, clustering words that have similar life-cycles and, thus, detecting prototypical or exemplary temporal patterns. Unveiling such relevant and (through expert opinion) meaningful inner dynamics enables us to trace a historical narrative of the discipline of interest. However, different history readings are possible depending on the type of data normalization, which is needed to account for the fluctuating size of texts across time and the general problems of data sparsity and strong asymmetry. This study proposes a methodology consisting of (1) a stepwise information retrieval procedure for keywords’ selection and (2) a functional clustering two-stage approach for statistical learning. Moreover, a sample of possible normalizations of word frequencies is considered, showing that the different concept of curve similarity induced in clustering by the type of transformation heavily affects groups’ composition and size. The corpus of titles of scientific papers published by the American Statistical Association journals in the time span 1888–2012 is examined for illustration. [ABSTRACT FROM AUTHOR]

Many data collected in sport science come from time dependent phenomenon. This article focuses on Functional Data Analysis (FDA), which study longitudinal data by modelling them as continuous functions. After a brief review of several FDA methods, some useful practical tools such as Functional Principal Component Analysis (FPCA) or functional clustering algorithms are presented and compared on simulated data. Finally, the problem of the detection of promising young swimmers is addressed through a curve clustering procedure on a real data set of performance progression curves. This study reveals that the fastest improvement of young swimmers generally appears before 16 years old. Moreover, several patterns of improvement are identified and the functional clustering procedure provides a useful detection tool.

Air traffic management (ATM) aims at providing companies with a safe and ideally optimal aircraft trajectory planning. Air traffic controllers act on flight paths in such a way that no pair of aircraft come closer than the regulatory separation norms. With the increase of traffic, it is expected that the system will reach its limits in the near future: a paradigm change in ATM is planned with the introduction of trajectory-based operations. In this context, sets of well-separated flight paths are computed in advance, tremendously reducing the number of unsafe situations that must be dealt with by controllers. Unfortunately, automated tools used to generate such planning generally issue trajectories not complying with operational practices or even flight dynamics. In this paper, a means of producing realistic air routes from the output of an automated trajectory design tool is investigated. For that purpose, the entropy of a system of curves is first defined, and a mean of iteratively minimizing it is presented. The resulting curves form a route network that is suitable for use in a semi-automated ATM system with human in the loop. The tool introduced in this work is quite versatile and may be applied also to unsupervised classification of curves: an example is given for French traffic. [ABSTRACT FROM AUTHOR]

We propose a new unsupervised learning algorithm to fit regression mixture models with unknown number of components. The developed approach consists in a penalized maximum likelihood estimation carried out by a robust expectation–maximization (EM)-like algorithm. We derive it for polynomial, spline, and B-spline regression mixtures. The proposed learning approach is unsupervised: (i) it simultaneously infers the model parameters and the optimal number of the regression mixture components from the data as the learning proceeds, rather than in a two-fold scheme as in standard model-based clustering using afterward model selection criteria, and (ii) it does not require accurate initialization unlike the standard EM for regression mixtures. The developed approach is applied to curve clustering problems. Numerical experiments on simulated and real data show that the proposed algorithm performs well and provides accurate clustering results, and confirm its benefit for practical applications. [ABSTRACT FROM PUBLISHER]

Deformations in tunnels and galleries due to confinement pressures can occur continuously but also at discrete time intervals. We propose a methodology to detect discrete significant deformations in tunnels based on a probabilistic model-based curve clustering. An EM (expectation-maximization) algorithm is used to obtain the parameters of the component density functions that maximize the log-likelihood function. The estimation of the number of clusters was performed by means of the Bayesian Information Criterion (BIC). The proposed methodology was applied to the analysis of the deformations in a tunnel that has been used in the past to transport coal in an underground mine. A set of 40 profiles measured over a period of 20 months were compared. The results obtained show that deformations are not continuous but significantly high deformation episodes occur. [ABSTRACT FROM AUTHOR]

Air traffic management (ATM) aims at providing companies with a safe and ideally optimal aircraft trajectory planning. Air traffic controllers act on flight paths in such a way that no pair of aircraft come closer than the regulatory separation norms. With the increase of traffic, it is expected that the system will reach its limits in the near future: a paradigm change in ATM is planned with the introduction of trajectory-based operations. In this context, sets of well-separated flight paths are computed in advance, tremendously reducing the number of unsafe situations that must be dealt with by controllers. Unfortunately, automated tools used to generate such planning generally issue trajectories not complying with operational practices or even flight dynamics. In this paper, a means of producing realistic air routes from the output of an automated trajectory design tool is investigated. For that purpose, the entropy of a system of curves is first defined, and a mean of iteratively minimizing it is presented. The resulting curves form a route network that is suitable for use in a semi-automated ATM system with human in the loop. The tool introduced in this work is quite versatile and may be applied also to unsupervised classification of curves: an example is given for French traffic.

Tracer kinetic modeling with dynamic positron emission tomography (PET) requires a plasma time-activity curve (PTAC) as an input function. Several image-derived input function (IDIF) methods that rely on drawing the region of interest (ROI) in large vascular structures have been proposed to overcome the problems caused by the invasive approach for obtaining the PTAC, especially for small-animal studies. However, the manual placement of ROIs for estimating IDIF is subjective and labor-intensive, making it an undesirable and unreliable process. In this paper, we propose a novel hybrid clustering method (HCM) that objectively delineates ROIs in dynamic PET images for the estimation of IDIFs, and demonstrate its application to the mouse PET studies acquired with [ ^18F]Fluoro-2-deoxy-2-D-glucose (FDG). We begin our HCM using k-means clustering for background removal. We then model the time-activity curves using polynomial regression mixture models in curve clustering for heart structure detection. The hierarchical clustering is finally applied for ROI refinements. The HCM achieved accurate ROI delineation in both computer simulations and experimental mouse studies. In the mouse studies, the predicted IDIF had a high correlation with the gold standard, the PTAC derived from the invasive blood samples. The results indicate that the proposed HCM has a great potential in ROI delineation for automatic estimation of IDIF in dynamic FDG-PET studies. [ABSTRACT FROM AUTHOR]

This brief presents a curve clustering technique based on a new multivariate model. Instead of the usual Gaussian random effect model, our method uses the multivariate t-distribution model which has better robustness to outliers and noise. In our method, we use the B-spline curve to model curve data and apply the mixed-effects model to capture the randomness and covariance of all curves within the same cluster. After fitting the B-spline-based mixed-effects model to the proposed multivariate t-distribution, we derive an expectation-maximization algorithm for estimating the parameters of the model, and apply the proposed approach to the simulated data and the real dataset. The experimental results show that our model yields better clustering results when compared to the conventional Gaussian random effect model. [ABSTRACT FROM AUTHOR]

We present a new approach for curve clustering designed for analysis of spatiotemporal data. Such data contains both spatial and temporal patterns that we desire to capture. The proposed methodology is based on regression and Gaussian mixture modeling. The novelty of the herein work is the incorporation of spatial smoothness constraints in the form of a prior for the data labels. This allows to take into account the property of spatiotemporal data according to which spatially adjacent data points have higher probability to belong to the same cluster. The proposed model can be formulated as a Maximum a Posteriori (MAP) problem, where the Expectation Maximization (EM) algorithm is used to estimate the model parameters. Several numerical experiments with both simulated data and real cardiac perfusion MRI data are used for evaluating the methodology. The results are promising and demonstrate the value of the proposed approach. [ABSTRACT FROM AUTHOR]

In recent years, in many application fields, extracting information from data in the form of functions is of most interest rather than investigating traditional multivariate vectors. Often these functions have complex spatial dependences that need to be accounted for using appropriate statistical analysis. Spatial Functional Statistics presents a fruitful analytics framework for this analysis. The definition of a distance measure between spatially dependent functional data is critical for many functional data analysis tasks such as clustering and classification. For this reason, and based on the specific characteristics of functional data, several distance measures have been proposed in the last few years. In this work we develop a weighted L 2 distance for spatially dependent functional data, with an optimized weight function. Assuming a penalized basis representation for the functional data, we consider weight functions depending also on the spatial location in two different situations: a classical georeferenced spatial structure and a connected network one. The performance of the proposed distances are compared using standard metrics applied to both real and simulated data analysis. [ABSTRACT FROM AUTHOR]